Question

. Find the inverse of the function below on the given interval and write it in the form yequalsf Superscript negative 1 Baseline (x ). b. Verify the relationships f (f Superscript negative 1 Baseline (x ))equalsx and f Superscript negative 1 Baseline (f (x ))equalsx.

Answer 1

The inverse of the function is
`f^(-1)(x)=(x-5)/(3)`.

Explanation:

The function provided is:

`f (x)=3x+5`

Let
`f(x)=y`.

Then the value of x is:

`y=3x+5\\\\3x=y-5\\\\x=(y-5)/(3)`

For the inverse of the function,
`x\rightarrow y`.

⇒
`f^(-1)(x)=(x-5)/(3)`

Compute the value of
`f[f^(-1)(x)]` as follows:

`f[f^(-1)(x)]=f[(x-5)/(3)]`

`=3[(x-5)/(3)]+5\\\\=x-5+5\\\\=x`

Hence proved that
`f[f^(-1)(x)]=x`.

Compute the value of
`f^(-1)[f(x)]` as follows:

`f^(-1)[f(x)]=f^(-1)[3x+5]`

`=((3x+5)-5)/(3)\\\\=(3x+5-5)/(3)\\\\=x`

Hence proved that
`f^(-1)[f(x)]=x`.